Estimates of Hilbert modular cusp forms of half-integral and integral weight

TL;DR

This paper provides asymptotic estimates for sums of squared absolute values of Hilbert modular cusp forms of varying weights, extending results to integral weights, specific subgroups, and adelic forms, with bounds independent of the subgroup.

Contribution

It introduces new bounds for sums of cusp form values as weight increases, including cases for integral weights and specific arithmetic subgroups, broadening understanding of their growth behavior.

Findings

Sum of cusp form squares is bounded by O(k^n) as weight k increases.

Results extend to integral weights and subgroups commensurable with Hilbert modular groups.

Bounds are independent of the subgroup mma.

Abstract

Let $\Gamma$ be a cocompact, discrete, and irreducible subgroup of $\mathrm{PSL}_{2}(\mathbb{R})^{n}$. Let $\nu$ be a unitary character of $\Gamma$. For $k\in1\slash 2\,\mathbb{Z}$, let $\sknu$ denote the complex vector space of cusp forms of weight-$\tk=\k$ and nebentypus $\nu^{2k}$ with respect to $\Gamma$. We assume that $\omega_{X,\nu}$, the line bundle of cusp forms of weight-$\tilde{1\slash 2}:=(1\slash 2,\ldots,1\slash2)$ with nebentypus $\nu$ over $X$ exists. Let $\lbrace f_{1},\ldots,f_{j_{\tk}} \rbrace$ denote an orthonormal basis of $\sknu$. In this article, we show that as $k\rightarrow \infty$, the sum $\sum_{i=1}^{j_{\tk}}y^{k}|f_{i}(z)|^{2}$ is bounded by $O(k^{n})$, where the implied constant is independent of $\Gamma$. Furthermore, we extend these results to the case when $k\in2\mathbb{Z}$, and to the case when $\Gamma$ is commensurable with the Hilbert modular group $\Gamma_{K}:=\mathrm{PSL}_{2}(O_{K})$, where $K$ is a totally real number field of degree $n\geq 2$, and $\mathcal{O}_{K}$ is the ring of integers of $K$, and to the case of adelic modular forms.